Combinatorial Euclidean geometry problem

Question

Let $\mathcal{S}^d_{\epsilon}$ be the collection of all sets $S:=\{\mathbf{x}_1, \mathbf{x}_2, \ldots \mathbf{x}_{d+1}\}$ of $d+1$ points in a $d$-dimensional Euclidean space such that, for a given constant $\epsilon>0$, we have $\|\mathbf{x}_i-\mathbf{x}_j\|_2\in[1-\epsilon,1]$ for all $i\neq j$.

Given any set $S\in\mathcal{S}^d_{\epsilon}$, let $n(S)$ be the number of triangles whose vertices $\mathbf{x}_i, \mathbf{x}_j, \mathbf{x}_k$ are points of $S$, such that $\|\mathbf{x}_i-\mathbf{x}_j\|_2=\|\mathbf{x}_j-\mathbf{x}_k\|_2=1$ and $\|\mathbf{x}_k-\mathbf{x}_i\|_2<1$.

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Question: How can we calculate (or approximate) $m(d,\epsilon):=\max_{S\in\mathcal{S}^d_{\epsilon}}n(S)$?

Answer 1

Since any graph with $d+1$ vertices can be realized as a unit distance graph in $\mathbb{R}^d$, with remaining distances smaller than $1$ (and arbitrarily close to 1), the question is then equivalent to the maximum possible number of induced paths of length $2$ (equivalently, induced copies of $K_{2,1}$), in a graph with $d+1$ vertices.

This is maximized for the balanced complete bipartite graph, where approximately half of the triples of vertices induce $K_{2,1}$; see Theorem 10 in [1]:

[1] N. Pippenger and M. C. Golumbic, The inducibility of graphs, Journal of Combinatorial Theory, Series B, Volume 19, Issue 3, December 1975, Pages 189-203; https://www.sciencedirect.com/science/article/pii/0095895675900842

In particular, $m(d,\epsilon)$ does not depend on $\epsilon$.